Coef encoding vs CRT encoding

Library Questions
1

Hello.

I have been watching webinars Homomorphic Encryption for OpenFHE Users – OpenFHE.org and reading forum posts such as Ressources for Batching, but I am wondering if my understanding is correct.

For coef encoding, e.g. MakeCoefPackedPlaintext, the elements of the vector are directly coefficients of a polynoimal in the plaintext ring R_t for plaintext modulus t. Addition can be done without issue since each coefficient can be handled separetly.

Meanwhile, for CRT encoding, e.g. MakePackedPlaintext, the elements of the vector are instead encoded as elements in different polynomial rings R_{t_1}, R_{t_2}, \dots for which the product is isomorphic to the ring R_t. This allows multiplication to be performed, but requires that t satisfy t = 1 (mod 2n) for ring dimension / vector length n.

Is this understanding correct? Also, if I only require addition should I always use coef encoding or are there situations where CRT encoding is still better?

2

Your understanding is correct. The only error is that even after CRT encoding, each element is in \mathbb{Z}_t (there are no t_1 and t_2). The product of all of them corresponds to the full encoding.

Yes, this is true for power-of-two cyclotomics, i.e., when n is a power of two (which is the setting used by OpenFHE).

Yes, in the case of addition only coefficient encoding is better because no special constraints for t are needed. For example, you can use t=2 even for the power-of-two-cyclotomics setting.

3

Thank you for the answer.